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This paper concerns the modeling of multi-way functional data where double ormultiple indices are involved. We introduce a concept of weak separability. Theweakly separable structure supports the use of factorization methods thatdecompose the signal into its spatial and temporal components. The analysisreveals interesting connections to the usual strongly separable covariancestructure, and provides insights into tensor methods for multi-way functionaldata. We propose a formal test for the weak separability hypothesis, where theasymptotic null distribution of the test statistic is a chi-square typemixture. The method is applied to study brain functional connectivity derivedfrom source localized magnetoencephalography signals during motor tasks.

A test of weak separability for multi-way functional data, with application to brain connectivity studies